I know there must be something unmathematical in the following but I don't know where it is: −1−−−√1−1−−−√1√−1−−−√1−1−−−√−11−−−√−1−−−√ii2−1=i=1i=1i=1i=1i=1i=1i=1=1!!!

Between your third and fourth lines, you use ab=ab. This is only (guaranteed to be) true when a0 and b>0.
edit: As pointed out in the comments, what I meant was that the identity ab=ab has domain a0 and b>0. Outside that domain, applying the identity is inappropriate, whether or not it "works."
In general (and this is the crux of most "fake" proofs involving square roots of negative numbers), x where x is a negative real number (x<0) must first be rewritten as i|x| before any other algebraic manipulations can be applied (because the identities relating to manipulation of square roots [perhaps exponentiation with non-integer exponents in general] require nonnegative numbers).
This similar question, focused on 1=i2=(1)2=11=!11=1=1, is using the similar identity ab=ab, which has domain a0 and b0, so applying it when a=b=1 is invalid.


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